Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle LOM = 2x + 38$, and $ m \angle MON = 9x - 58$, find $m\angle LOM$. $O$ $L$ $N$ $M$
Explanation: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {2x + 38} + {9x - 58} = {90}$ Combine like terms: $ 11x - 20 = 90$ Add $20$ to both sides: $ 11x = 110$ Divide both sides by $11$ to find $x$ $ x = 10$ Substitute $10$ for $x$ in the expression that was given for $m\angle LOM$ $ m\angle LOM = 2({10}) + 38$ Simplify: $ {m\angle LOM = 20 + 38}$ So ${m\angle LOM = 58}$.